Geometric characterization of intermittency in the parabolic Anderson model

Abstract

We consider the parabolic Anderson problem ∂_tu=Δu+ξ(x)u on ℝ₊×ℤ^d with localized initial condition u(0, x)=δ₀(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑_xu(t, x) comes from a slowly increasing number of “islands” which are located far from each other. These “islands” are local regions of those high exceedances of the field ξ in a box of side length 2t log²t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.